I've just had a new installation of a "BT" line as I was with virgin before. It is still a copper pair from the cabinet to the house but no traditional phone just ADSL and voip from a socket on the router.
I can't see how BT will achieve a connection during power cuts as it would not only require a battery backup in the home but also to the box in the street with giga bits of data to handle. In essence more bandwidth needs more power and half the cabinet would have to be batteries.
Even if the phone back up were to be via a mobile mast, BT might have done their bit but no guarantee of the local mast having mains power.